Hierarchy of the Binary Models r=nr=nk=rk=r k-out-of-r-from-n:F r n Consecutive k-out-of-n k n n k-out-of-n:F.

Hierarchy of the Binary Models

r=n k=r

k-out-of-r-from-n:Fr

n

Consecutive k-out-of-nk

nn

k-out-of-n:F

0 gnom

1

r

Pr{g>x}

x 0 gn

1

r

Pr{g>x}

xg1 g2 ...

Binary element Multi-state element

Multi-state Models

k-out-of-n

Weighted k-out-of-nWu, Chen (1994)

Parallel Multi-state System

Multi-stateconsecutive k-out-of-n

Hwang, Yao (1989),Kossow, Preuss (1995)

Consecutive k-out-of-n

Chiang, Niu (1981),Bollinger (1982)

Sliding WindowSystems

Levitin (2002)

k-out-of-r-from-nGriffith (1986)

r=n k=r

r

k-out-of-r-from-n:

}1,0{

11,1

m

rh

hmm

G

rnhkG

Sliding window system definition

1

111 ),...,(),...,(

rn

hrhhn GGfGGF

Acceptability function

Any function of r variables Any real value

Total number of groups: n-r+1

...

Each element belongs to no more than r groups

...

Sliding window systems

SWS Applications: Manufacturing

nr

r

n

SWS Applications: Service System

SWS Applications: Quality Control

n

r

Deviation Levels

3 2 1 0 1 2 3

Cyclic Buffer

gi,k

gi+1,kgi+2,k

gi+r-1,k

...

...

i

k,iK

1k

Ok,ii zp)z(

Element State Distribution

r-Group State Distribution

i

j,iN

1j

gj,ii zq)z(u

Representing Multi-state Elements and Groups

gi,k

gi+1,kgi+2,k

gi+r-1,k

gi+r,j+gi+r,k-gi,j

...

...

Composition Operator

rij,rik,i

i N

1j

gOj,rik,i

K

1krii1i zqp)z(u)z()z(

Operator for Determining Group Unreliability

iK

1kk,ik,ii ).w)O((1p))z((

gi

gi+1gi+2

gi+r-1

gi+r,j

...

...

)(1

,,

1

,,

Ni

kki

kiOm

k

kiOki pzzp

Like term collection in the the u-function

g i+r-1 g i+r-1

gi,1 gi,2 gi,3 gi,Ni

...

Algorithm for SWS Reliability Determination

1. Initialization

F=0; 1-r(z) = 0z .

Determine u-functions of the individual MEs uj(z).

2. Main loop

Repeat the following for j=1,…,n:

2.1. Obtain (z)ju(z)rjΨ(z)r1jΨ .

2.2. If jr add value W)(z),r1jδ(Ψ to F

and remove all the terms with <W from (z)r1jΨ

3. Obtain the SWS reliability as R=1-F.

0

0.2

0.4

0.6

0.8

1

0 3 6 9 12 15 18 21 24 27 30W

R

2 3 4 5 6 7 8 9 10r:

0

0.3

0.6

0.9

0 1 2 3 4x

P{G>x)

Element performance distribution

Example of SWS reliability Determination

10 identical elements

Reliability Importance of SWS Elements

0

0.3

0.6

0.9

0 100 200 300 400 500 600

1 2 3 4 5

6 7 8 9 10

No 1 2 3 4 5 6 7 8 9 10

r 0.87 0.90 0.83 0.95 0.92 0.89 0.80 0.85 0.82 0.95

g 200 200 400 300 100 400 100 200 300 200

Irrelevant element

Most important

element

Ij= R/ rj

I

w

Optimal Sequencing of SWS Elements

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16

optimal for w=6 optimal for w=8 optimal for w=10

R

w

1 2 3 4 5p g p g p g p g p g

0.03 0 0.1 0 0.17 0 0.05 0 0.08 00.22 2 0.1 1 0.83 6 0.25 3 0.2 10.75 5 0.4 2 - - 0.4 5 0.15 2

- - 0.4 4 - - 0.3 6 0.45 4- - - - - - - - 0.12 5

6 7 8 9 10p g p g p g p g p g

0.01 0 0.2 0 0.05 0 0.2 0 0.05 00.22 4 0.1 3 0.25 4 0.1 3 0.25 20.77 5 0.1 4 0.7 6 0.15 4 0.7 6

- - 0.6 5 - - 0.55 5 - -

2,1,6,5,4,8,7,10,3,9 5,1,8,9,6,4,7,3,10,2 5,9,3,1,4,7,10,8,6,2

SWS Elements Performance distribution

SWS Reliability

A

B

RA(3) = p4; RA(4) =0

RB(3) = p4+4(1-p)p3; RB(4) = p4

5—9—3—1—4—7—10—8—6—2

— —6,7,10— —2,5—1,4— —3,8,9— —

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

W

R

Uneven allocation of SWS elements

Optimal Grouping of SWS Elements in the Presence of Common Cause Failures

r=3 r=5

0

0.2

0.4

0.6

0.8

1

0 5 10 15

w

R

M=1 M=2 M=3

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25w

R

M=1 M=2 M=3

M=4 M=5

Optimal Grouping Solutions for Different r and M

r=3 r=5

0

0.1

0.2

0.3

0.4

0.5

0 5 10 15w

I

CSG1 CSG2 CSG3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20 25w

I

CSG1 CSG2 CSG3

Group Survivability Importance

Ij= R/ sj

r1=2, w1

r2=6, w2

r3=3, w3

g1 g2 g3 g4

Multiple sliding window systems

r1=3r2=5

G1 … …Gn

012345

01

23

0

0.25

0.5

0.75

1R

w2w1

w3=0

012

34

5

01

23

0

0.25

0.5

0.75

1R

w2w1

w3=5

01234

5

01

23

0

0.25

0.5

0.75

1 R

w2w1

w3=6

>w1>w2

>w3

}1.0{

Example of SMWS